Write a possible exponential function in y = a · bx form that represents each situation described below. Homework Help ✎Has a y‑intercept of (0, 2) and passes through the point (3, 128). Passes through the points (0, 4) and (2, 1).
Accepted Solution
A:
Answer:[tex]y=2\cdot4^x\\y=4\cdot\left(\frac{1}{2}\right)^x[/tex]Step-by-step explanation:We're given both the y-intercept and a point on the graphs of both functions, so our work her is largely just substituting x and y values in the general form of the equation for an exponential function.For the first one, we can start by using the point (0, 2) to solve for a:[tex]y=a\cdot b^x\\2=a\cdot b^0\\2=a[/tex]Next, we can use that a-value and the second point to solve for b:[tex]y=2\cdot b^x\\128=2\cdot b^3\\64=b^3\\4=b[/tex]This gives us the equation [tex]y=2\cdot4^x[/tex] for the first function.We can repeat the same process for the second function. Solving for a:[tex]y=a\cdot b^x\\4=a\cdot b^0\\4=a[/tex]And then for b, using the point (2, 1):[tex]1=4\cdot b^2\\\frac{1}{4}=b^2\\\frac{1}{2}=b[/tex]This gives us the general equation [tex]y=4\cdot\left(\frac{1}{2}\right)^x[/tex]